Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
Algebraic Geometry
Ivan Tomašic
Queen Mary, University of London
LTCC 2012
1 / 142
Outline
Varieties and schemesAffine varietiesSheavesSchemesProjective varietiesFirst properties of schemes
Local propertiesNonsingular schemesDivisorsRiemann-Roch Theorem
Weil conjectures
2 / 142
What is algebraic geometry?
IntuitionAlgebraic geometry is the study of geometric shapes that canbe (locally/piecewise) described by polynomial equations.
Why restrict to polynomials?
Because they make sense in any field or ring, including theones which carry no intrinsic topology.This gives a ‘universal’ geometric intuition in areas whereclassical geometry and topology fail.Applications in number theory: Diophantine geometry.Even in positive characteristic.
3 / 142
Example
A plane curve X defined by
x2 + y2 − 1 = 0.
I Over R, this defines a circle.I Over C, it is again a quadratic curve, even though it may be
difficult to imagine (as the complex plane has realdimension 4).
4 / 142
k -valued points
But we can consider the solutions
X (k) = (x , y) ∈ k2 : x2 + y2 = 1
for any field k .
I What can be said about X (Q)? It is infinite, think ofPythagorean triples, e.g. (3/5,4/5) ∈ X (Q).
I How about X (Fq)? With certainty we can say
|X (Fq)| < q · q = q2,
but this is a very crude bound. We intend to return to thisissue (Weil conjectures/Riemann hypothesis for varietiesover finite fields) at the end of the course.
5 / 142
Problems with non-algebraically closed fields
Example
Problem: for a plane curve Y defined by x2 + y2 + 1 = 0,
Y (R) = ∅.
6 / 142
(Historical) approaches
I Thus, if we intend to pursue the line of naïve algebraicgeometry and study algebraic varieties through their setsof points, we better work over an algebraically closed field.
I Italian school: Castelnuovo, Enriques, Severi–intuitiveapproach, classification of algebraic surfaces;
I American school: Chow, Weil, Zariski–gave solid algebraicfoundation to above.
I For the scheme-theoretic approach, we can work overarbitrary fields/rings, and the machinery of schemesautomatically performs all the necessary bookkeeping.
I French school: Artin, Serre, Grothendieck–schemes andcohomology.
7 / 142
Affine space
DefinitionLet k be an algebraically closed field.I The affine n-space is
Ank = (a1, . . . ,an) : ai ∈ k.
I LetA = k [x1, . . . , xn]
be the polynomial ring in n variables over k .I Think of an f ∈ A as a function
f : Ank → k ;
for P = (a1, . . . ,an) ∈ An, we let f (P) = f (a1, . . . ,an).
8 / 142
Vanishing set
Definition
I For f ∈ A, we let
V (f ) = P ∈ An : f (P) = 0.
I LetD(f ) = An \ V (f ).
I More generally, for any subset E ⊆ A,
V (E) = P ∈ An : f (P) = 0 for all f ∈ E =⋂f∈E
V (f ).
9 / 142
Properties of V
Proposition ()
I V (0) = An, V (1) = ∅;I E ⊆ E ′ implies V (E) ⊇ V (E ′);I for a family (Eλ)λ, V (∪λEλ) = V (
∑λ Eλ) = ∩λV (Eλ);
I V (EE ′) = V (E) ∪ V (E ′);I V (E) = V (
√〈E〉), where 〈E〉 is an ideal of A generated by
E and√· denotes the radical of an ideal,√
I = a ∈ A : an ∈ I for some n ∈ N.
This shows that sets of the form V (E) for E ⊆ A (calledalgebraic sets) are closed sets of a topology on An, which wecall the Zariski topology.Note: D(f ) are basic open.
10 / 142
Example
Algebraic subsets of A1 are just finite sets.
Thus any two open subsets intersect, far from being Hausdorff.
Proof.A = k [x ] is a principal ideal domain, so every ideal a in A isprincipal, a = (f ), for f ∈ A. Since k is ACF, f splits in k , i.e.
f (x) = c(x − a1) · · · (x − an).
Thus V (a) = V (f ) = a1, . . . ,an.
11 / 142
Affine varieties
DefinitionAn affine algebraic variety is a closed subset of An, togetherwith the induced Zariski topology.
12 / 142
Associated Ideal
DefinitionLet Y ⊆ An be an arbitrary set (not necessarily closed). Theideal of Y in A is
I(Y ) = f ∈ A : f (P) = 0 for all P ∈ Y.
Proposition
1. Y ⊆ Y ′ implies I(Y ) ⊇ I(Y ′);2. I(∪λYλ) = ∩λI(Yλ);3. for any Y ⊆ An, V (I(Y )) = Y, the Zariski closure of Y in
An;4. for any E ⊆ A, I(V (E)) =
√〈E〉.
13 / 142
Proof.3. Clearly, V (I(Y )) is closed and contains Y . Conversely, ifV (E) ⊇ Y , then, for every f ∈ E , f (y) = 0 for every y ∈ Y , sof ∈ I(Y ), thus E ⊆ I(Y ) and V (E) ⊇ V (I(Y )).
4. Is commonly known as Hilbert’s Nullstellensatz. Let us writea = 〈E〉. It is clear that
√a ⊆ I(V (a)). For the converse
inclusion, we shall assume:
the weak Nullstellensatz (in (n + 1) variables):
for a proper ideal J in k [x0, . . . , xn], we have V (J) 6= 0 (it iscrucial here that k is algebraically closed).
Suppose f ∈ I(V (a)). The ideal J = 〈1− x0f 〉+ a ink [x0, . . . , kn] has no zero in kn+1 so we conclude J = 〈1〉, i.e.1 ∈ J. It follows (by substituting 1/f for x0 and clearingdenominators) that f n ∈ a for some n.For a complete proof see Atiyah-Macdonald.
14 / 142
Quasi-compactness
Corollary
D(f ) is quasi-compact. (not Hausdorff )
Proof.If ∪iD(fi) = D(f ), then V (f ) = ∩iV (fi) = V (fi : i ∈ I), sof ∈
√fi : i ∈ I, so there is a finite I0 ⊆ I with
f ∈√fi : i ∈ I0.
15 / 142
Corollary
There is a 1-1 inclusion-reversing correspondence
Y 7−→ I(Y )
V (a)←− [ a
between algebraic sets and radical ideals.
16 / 142
Given a point P = (a1, . . . ,an) ∈ An, the ideal mP = I(P) ismaximal (because the set P is minimal), andmP = (x1 − a1, . . . , xn − an). Weak Nullstellensatz tells us thatevery maximal ideal is of this form.Thus,
I(V (a)) =⋂
P∈V (a)
I(P) =⋂
P∈V (a)
mP =⋂m⊇a
m maximal
m.
On the other hand, it is known in commutative algebra that
√a =
⋂p⊇a
p prime
p
Thus, Nullstellensatz in fact claims that the two intersectionscoincide, i.e., that A is a Jacobson ring.
17 / 142
Affine coordinate ring
DefinitionIf Y is an affine variety, its affine coordinate ring isO(Y ) = A/I(Y ).
O(Y ) should be thought of as the ring of polynomial functionsY → k . Indeed, two polynomials f , f ′ ∈ A define the samefunction on Y iff f − f ′ ∈ I(Y ).
18 / 142
Remark
I If Y is an affine variety, O(Y ) is a finitely generatedk-algebra.
I Conversely, any finitely generated reduced (no nilpotentelements) k-algebra is a coordinate ring of an irreducibleaffine variety.
Indeed, suppose B is generated by b1, . . . ,bn as a k -algebra,and define a morphism A = k [x1, . . . , xn]→ B by xi 7→ bi . SinceB is reduced, the kernel is a radical ideal a, so B = O(V (a)).
19 / 142
Maximal spectrum
RemarkLet Specm(B) denote the set of all maximal ideals of B. Thenwe have 1-1 correspondences between the following sets:
1. (points of) Y ;2. Y (k) := Homk (O(Y ), k);3. Specm(O(Y ));4. maximal ideals in A containing I(Y ).
Let P ∈ Y , P = (a1, . . . ,an). We know I(P) ⊇ I(Y ), so themorphism a : O(Y ) = A/I(Y )→ k , xi + I(Y ) 7→ ai iswell-defined. Since the range is a field, mP = ker(a) is maximalin O(Y ), and its preimage in A is exactlyI(P) = f ∈ A : f (P) = 0.
20 / 142
Irreducibility
DefinitionA topological space X is irreducible if it cannot be written as theunion X = X1 ∪ X2 of two proper closed subsets.
Proposition
An algebraic variety is irreducible iff its ideal is prime iff O(Y ) isa domain.
21 / 142
Proof.Suppose Y is irreducible, and let fg ∈ I(Y ). Then
Y ⊆ V (fg) = V (f ) ∪ V (g) = (Y ∩ V (f )) ∪ (Y ∩ V (g)),
both being closed subsets of Y . Since Y is irreducible, we haveY = Y ∩V (f ) or Y = Y ∩V (g), i.e., Y ⊆ V (f ) or Y ⊆ V (g), i.e.,f ∈ I(Y ) or g ∈ I(Y ). Thus I(Y ) is prime.Conversely, let p be a prime ideal and suppose V (p) = Y1 ∪ Y2.Then p = I(Y1) ∩ I(Y2) ⊇ I(Y1)I(Y2), so we have p = I(Y1) orp = I(Y2), i.e., Y1 = V (p) or Y2 = V (p), and we conclude thatV (p) is irreducible.
22 / 142
Examples
I An is irreducible; An = V (0) and 0 is a prime ideal since Ais a domain.
I if P = (a1, . . . ,an) ∈ An, then P = V (mP),mP = (x1 − a1, . . . , xn − an) is a max ideal, hence prime, soP is irreducible.
I Let f ∈ A = k [x , y ] be an irreducible polynomial. Then V (f )is an irreducible variety (affine curve); (f ) is prime since Ais an unique factorisation domain.
I V (x1x2) = V (x1) ∪ V (x2) is connected but not irreducible.
23 / 142
Noetherian topological spaces
DefinitionA topological space X is noetherian, if it has the descendingchain condition (or DCC) on closed subsets: any descendingsequence Y1 ⊇ Y2 ⊇ · · · of closed subsets eventuallystabilises, i.e., there is an r ∈ N such that Yr = Yr+i for all i ∈ N.
Proposition ()
In a noetherian topological space X, every nonempty closedsubset Y can be expressed as an irredundant finite union
Y = Y1 ∪ · · · ∪ Yn
of irreducible closed subsets Yi (irredundant means Yi 6⊆ Yj fori 6= j ).The Yi are uniquely determined, and we call them theirreducible components of Y .
24 / 142
Noetherian rings
DefinitionA ring A is noetherian if it satisfies the following threeequivalent conditions:
1. A has the ascending chain condition on ideals: everyascending chain I1 ⊆ I2 ⊆ · · · of ideals is stationary(eventually stabilises);
2. every non-empty set of ideals in A has a maximal element;3. every ideal in A is finitely generated.
25 / 142
Hilbert’s Basis TheoremTheorem (Hilbert’s Basis Theorem)
If A is noetherian, then the polynomial ring A[x1, . . . , xn] isnoetherian.
Corollary
If A is noetherian and B is finitely generated A-algebra, then Bis also noetherian.
RemarkThis means that any algebraic variety Y ⊆ An is in fact a set ofsolutions of a finite system of polynomial equations:
f1(x1, . . . , xn) = 0...
fm(x1, . . . , xn) = 0
26 / 142
Irreducible components
Corollary
Every affine algebraic variety is a noetherian topological spaceand can be expressed uniquely as an irredundant union ofirreducible varieties.
Proof.O(Y ) is a finitely generated k -algebra and a field k is triviallynoetherian, so O(Y ) is a noetherian ring. A descending chainof closed subsetsY1 ⊇ Y2 ⊇ · · · in Y gives rise to an ascendingchain of ideals I(Y1) ⊆ I(Y2) ⊆ · · · in O(Y ), which must bestationary. Thus the original chain of closed subsets must bestationary too.
27 / 142
Finding/computing irreducible components in a concrete caseis a non-trivial task, which can be made efficient by the use ofGröbner bases.
Example (Exercise)
Let Y = V (x2 − yz, xz − x) ⊆ A3. Show that Y is a union of 3irreducible components and find their prime ideals.
28 / 142
Dimension
Definition
I The dimension of a topological space X is the supremumof all n such that there exists a chain
Z0 ⊂ Z1 ⊂ · · · ⊂ Zn
of distinct irreducible closed subsets of X .I The dimension of an affine variety is the dimension of its
underlying topological space.
not every noetherian space has finite dimension.
29 / 142
Definition
I In a ring A, the height of a prime ideal p is the supremum ofall n such that there exists a chain p0 ⊂ p1 ⊂ · · · ⊂ pn = pof distinct prime ideals.
I The Krull dimension of A is the supremum of the heights ofall the prime ideals.
FactLet B be a finitely generated k-algebra which is a domain. Then
1. dim(B) = tr.deg(k(B)/k), where k(B) is the fraction field ofB;
2. for any prime ideal p of B,
height(p) + dim(B/p) = dim(B).
30 / 142
Topological and algebraic dimension
Proposition
For an affine variety Y ,
dim(Y ) = dim(O(Y )).
By the previous Fact, the latter equals the number ofalgebraically independent coordinate functions, and we deduce:
Proposition
dim(An) = n.
31 / 142
Proposition ()
Let Y be an affine variety.1. If Y is irreducible and Z is a proper closed subset of Y ,
then dim(Z ) < dim(Y ).2. If f ∈ O(Y ) is not a zero divisor nor a unit, then
dim(V (f ) ∩ Y ) = dim(Y )− 1
Examples
1. Let X ,Y ⊆ A2 be two irreducible plane curves. Thendim(X ∩ Y ) < dim(X ) = 1, so X ∩ Y is of dimension 0 andthus it is a finite set.
2. A classification of irreducible closed subsets of A2.I If dim(Y ) = 2 = dim(A2), then by Prop, Y = A2;I If dim(Y ) = 1, then Y 6= A2 so 0 6= I(Y ) is prime and thus
contains a non-zero irreducible polynomial f . SinceY ⊇ V (f ) and dim(V (f )) = 1, it must be Y = V (f ).
I If dim(Y ) = 0, then Y is a point.32 / 142
Example (The twisted cubic curve)
Let Y ⊆ A3 be the set t , t2, t3) : t ∈ k. Show that it is anaffine variety of dimension 1 (i.e., an affine curve).Hint: Find the generators of I(Y ) and show that O(Y ) isisomorphic to a polynomial ring in one variable over k .
33 / 142
Morphisms of affine varieties
DefinitionLet X ⊆ An and Y ⊆ Am be two affine varieties. A morphism
ϕ : X → Y
is a map such that there exist polynomialsf1, . . . , fm ∈ k [x1, . . . , xn] with
ϕ(P) = (f1(a1, . . . ,an), . . . , fm(a1, . . . ,an)),
for every P = (a1, . . . ,an) ∈ X .
Remark ()
Morphisms are continuous in Zariski topology.
34 / 142
Morphisms vs algebra morphisms
A morphism ϕ : X → Y defines a k -homomorphism
ϕ : O(Y )→ O(X ), ϕ(g) = g ϕ,
when g ∈ O(Y ) is identified with a function Y → k .
A k -homomorphism ψ : O(Y )→ O(X ) defines a morphism
aψ : X → Y .Identify X with X (k) = Hom(O(X ), k) and Y by Y (k). Then
aψ(x) = x ψ.
Proposition a(ϕ) = ϕ and (aψ) = ψ.
35 / 142
Duality between algebra and geometry
Corollary
The functorX 7−→ O(X )
defines an arrow-reversing equivalence of categories ()between the category of affine varieties over k and thecategory of finitely generated reduced k-algebras.
I The ‘inverse’ functor is A 7→ Specm(A). For ψ : B → A,Specm(ψ) = aψ : Specm(A)→ Specm(B),aψ(m) = ψ−1(m), m a max ideal in A.
I This means that X and Y are isomorphic iff O(X ) andO(Y ) are isomorphic as k -algebras.
36 / 142
A translation mechanism
That means: every time you see a morphism
X −→ Y ,
you should be thinking that this comes from a morphism
O(X )←− O(Y ),
and vice-versa, every time you see a morphism
A←− B,
you should be thinking of a morphism
Specm(A) −→ Specm(B).
37 / 142
Methodology of algebraic geometry
I In physics, one often studies a system X by consideringcertain ‘observable’ functions on X .
I In algebraic geometry, all of the relevant information aboutan affine variety X is contained in its coordinate ring O(X ),and we can study the geometric properties of X by usingthe tools of commutative algebra on O(X ).
38 / 142
Examples ()
1. Let X = A1 and Y = V (x3 − y2) ⊆ A2, and let
ϕ : X → Y , defined by t 7→ (t2, t3).
Then ϕ is a morphism which is bijective and bicontinuous(a homeomorphism in Zariski topology), but ϕ is not anisomorphism.
2. Let char(k) = p > 0. The Frobenius morphism
ϕ : A1 → A1, t 7→ tp
is a bijective and bicontinuous morphism, but it is not anisomorphism.
39 / 142
Sheaves
DefinitionLet X be a topological space. A presheaf F of abelian groupson X consists of the data:I for every open set U ⊆ X , an abelian group F (U);
I for every inclusion Vi→ U of open subsets of X , a
morphism of abelian groups ρUV = F (i) : F (U)→ F (V ),
such that1. ρUU = F (id : U → U) = id : F (U)→ F (U);
2. if Wj→ V
i→ U, then F (i j) = F (j) F (i), i.e.,
ρUW = ρVW ρUV .
40 / 142
Presheaves as functors
The axioms above, in categorical terms, state that a presheafF on a topological space X , is nothing other than acontravariant functor from the category Top(X ) of open subsetswith inclusions to the category of abelian groups:
F : Top(X )op → Ab.
41 / 142
Sections jargon and stalks
For s ∈ F (U) and V ⊆ U, write s V = ρUV (s) and we refer toρUV as restrictions. Write (2) above as
s W = (s V ) W .
Elements of F (U) are sometimes called sections of F over U,and we sometimes write F (U) = Γ(U,F ), where Γ symbolises‘taking sections’.
DefinitionIf P ∈ X , the stalk FP of F at P is the direct limit of the groupsF (U), where U ranges over the open neighbourhoods of P (viathe restriction maps).
42 / 142
Stalks and germs of sections
Define the relation ∼ on pairs (U, s), where U is an open nhoodof P, and s ∈ F (U):
(U1, s1) ∼ (U2, s2)
if there is an open nhood W of P with W ⊆ U1 ∩ U2 such that
s1 W = s2 W .
Then FP equals the set of ∼-equivalence classes, which canbe thought of as ‘germs’ of sections at P.
43 / 142
Sheaves
DefinitionA presheaf F on a topological space X is a sheaf provided:
3. if Ui is an open covering of U, and s, t ∈ F (U) are suchthat s Ui = t Ui for all i , then s = t .
4. if Ui is an open covering of U, and si ∈ F (Ui) are suchthat for each i , j , si Ui∩Uj = sj Ui∩Uj , then there exists ans ∈ F (U) such that s Ui = si . (note that such an s isunique by 3.)
‘Unique glueing property’.
44 / 142
Examples
I Sheaf F of continuous R-valued functions on a topologicalspace X :
I F (U) is the set of continuous functions U → R,I for V ⊆ U, let ρUV : F (U)→ F (V ), ρUV (f ) = f V .
I Sheaf of differentiable functions on a differentiablemanifold;
I Sheaf of holomorphic functions on a complex manifold.I Constant presheaf: fix an abelian group Λ and let
F (U) = Λ for all U. This is not a sheaf (), its associatedsheaf satisfies
F +(U) = Λπ0(U),
where π0(U) is the number of connected components of U.(provided X is locally connected)
45 / 142
Sheaf morphisms
DefinitionLet F and G be presheaves of abelian groups on X .A morphism ϕ : F → G consists of the following data:I For each U open in X , we have a morphism
ϕ(U) : F (U)→ G (U).
I For each inclusion Vi→ U, we have a diagram
F (U) G (U)
F (V ) G (V )
ϕ(U)
F (i)ϕ(V )
G (i)
46 / 142
Sheaf morphisms as natural transformations
In categorical terms, if F and G are considered as functorsTop(X )op → Ab, a morphism
ϕ : F → G
is nothing other than a natural transformation () betweenthese functors.
47 / 142
Interlude on localisation
DefinitionLet A be a commutative ring with 1, and let S 3 1 be amultiplicatively closed subset of A. Define a relation ≡ on A×S:
(a1, s1) ≡ (a2, s2) if (a1s2 − a2s1)s = 0 for some s ∈ S.
Then ≡ is an equivalence relation and the ring of fractionsS−1A = A× S/ ≡ has the following structure (write a/s for theclass of (a, s)):
(a1/s1) + (a2/s2) = (a1s2 + a2s1)/s1s2,
(a1/s1)(a2/s2) = (a1a2/s1s2).
We have a morphism A→ S−1A, a 7→ a/1.
48 / 142
Interlude on localisation
Examples
I If A is a domain, S = A \ 0, then S−1A is the ring offractions of A.
I If p is a prime ideal in A, then S = A \ p is multiplicative andS−1A is denoted Ap and called the localisation of A at p.NBAp is indeed a local ring, i.e., it has a unique maximalideal.
I Let f ∈ A, S = f n : n ≥ 0. Write Af = S−1A.I S−1A = 0 iff 0 ∈ S.
49 / 142
Regular functions
RemarkLet X be an affine variety and g,h ∈ O(X ). Then
P 7→ g(P)
h(P)
is a well-defined function D(h)→ k.
We would like to consider functions defined on open subsets ofX which are locally of this form.
50 / 142
Regular functions
DefinitionLet U be an open subset of an affine variety X .I A function f : U → k is regular if for every P ∈ U,
there exist g,h ∈ O(X ) with h(P) 6= 0,and a neighbourhood V of P such thatthe functions f and g/h agree on V .
I The set of all regular functions on U is denoted OX (U).
Proposition ()
The assignment U 7→ OX (U) defines a sheaf of k-algebras onX.
It is called the structure sheaf of X .
51 / 142
Structure sheafProposition
Let X be an affine variety and let A = O(X ) be its coordinatering. Then:I For any P ∈ X, the stalk
OX ,P ' AmP ,
where the maximal ideal mP = f ∈ A : f (P) = 0 is theimage of I(P) in A.
I For any f ∈ A,OX (D(f )) ' Af .
I In particular,OX (X ) = A.
(so our notation for the coordinate ring is justified )52 / 142
Spectrum of a ringLet A be a commutative ring with 1.
DefinitionSpec(A) is the set of all prime ideals in A.
Our goal is to turn X = Spec(A) into a topological space andequip it with a sheaf of rings, i.e., make it into a ringed space.
Notation:
I write x ∈ X for a point, and jx for the corresponding primeideal in A;
I Ax = Ajx , the local ring at x ;I mx = jxAjx , the maximal ideal of Ax ;I k(x) = Ax/mx , the residue field at x , naturally isomorphic
to A/jx ;I for f ∈ A, write f (x) for the class of f mod jx in k(x). Then
‘f (x) = 0’ iff f ∈ jx .53 / 142
Examples
1. For a field F , Spec(F ) = 0, k(0) = F .2. Let Zp be the ring of p-adic integers. Spec(Zp) = 0, (p),
and k(0) = Qp, k((p)) = Fp. Generalises to an arbitraryDVR.
3. Spec(Z ) = 0 ∪ (p) : p prime . k(0) = Q, k((p)) = Fp.For f ∈ Z, f (0) = f/1 ∈ Q, and f (p) = f mod p ∈ Fp.
4. For an algebraically closed field k , let A = k [x , y ]. Then byClassification of irred subsets of A2
Spec(A) =0 ∪ (x − a, y − b) : a,b ∈ k∪ (g) : g ∈ A irreducible .
k(0) = k(x , y), k((x − a, y − b)) = k , k((g)) is the fractionfield of the domain A/(g). For f ∈ A, f (0) = f/1 ∈ k(x , y),f ((x − a, x − b)) = f (a,b) ∈ k , f ((g)) = (f + (g))/1 ∈ k(g).
54 / 142
Spectral topology
DefinitionFor f ∈ A, let
V (f ) = x ∈ X : f ∈ jx, i.e., the set of x with f (x) = 0;
D(f ) = X \ V (f ).
For E ⊆ A,
V (E) =⋂f∈E
V (f ) = x ∈ X : E ⊆ jx.
The operation V has expected properties: Jump to properties of V
Thus, the sets V (E) are closed sets for the Zariski topology onX .
55 / 142
DefinitionFor an arbitrary subset Y ⊆ X , the ideal of Y is
j(Y ) =⋂
x∈Y
jx i.e., the set of f ∈ A with f (x) = 0 for x ∈ Y ;
RemarkTrivially: √
E =⋂
x∈V (E)
jx .
The operation j has the expected properties: Jump to properties of I
and here the proof is trivial, no need for Nullstellensatz.
56 / 142
Direct image sheaf
DefinitionLet ϕ : X → Y be a continuous map of topological spaces andlet F be a presheaf on X . The direct image ϕ∗F is a presheafon Y defined by
ϕ∗F (U) = F (ϕ−1U).
Lemma ()
If F is a sheaf, so is ϕ∗F .
57 / 142
Ringed spacesDefinitionI A ringed space (X ,OX ) consists of a topological space X
and a sheaf of rings OX on X , called the structure sheaf.I A locally ringed space is a ringed space (X ,OX ) such that
every stalk OX ,x is a local ring, x ∈ X .I A morphism of ringed spaces (X ,OX )→ (Y ,OY ) is a pair
(ϕ, θ), where ϕ : X → Y is a continuous map, and
θ : OY → ϕ∗OX
is a map of structure sheaves.I (ϕ, θ) is a morphism of locally ringed spaces, if,
additionally, each induced map of stalks
θ]x : OY ,ϕ(x) → OX ,x
is a local homomorphism of local rings.58 / 142
Spectrum as a locally ringed space
LemmaThere exists a unique sheaf OX on X = Spec(A) satisfying
OX (D(f )) ' Af for f ∈ A.
Its stalks areOX ,x ' Ax (= Ajx ).
DefinitionBy Spec(A) we shall mean the locally ringed space
(Spec(A),OSpec(A)).
59 / 142
Schemes
Definition
I An affine scheme is a ringed space (X ,OX ) which isisomorphic to Spec(A) for some ring A.
I A scheme is a ringed space (X ,OX ) such that every pointhas an open affine neighbourhood U (i.e., (U,OX U) isan affine scheme).
I A morphism (X ,OX )→ (Y ,OY ) is just a morphism oflocally ringed spaces.
60 / 142
Ring homomorphisms induce morphisms of affineschemes
DefinitionA ring homomorphism ϕ : B → A gives rise to a morphism ofaffine schemes X = Spec(A) and Y = Spec(B):
(aϕ, ϕ) : (Spec(A),OSpec(A))→ (Spec(B),OSpec(B)),
whereI aϕ(x) = y iff jy = ϕ−1(jx ); (i.e., aϕ(p) = ϕ−1(p))I ϕ : OY → aϕ∗OX is characterised by (for g ∈ B):
OY (D(g)) OX (aϕ−1D(g)) OX (D(ϕ(g)))
Bg Aϕ(g)
ϕ(D(g))
b/gn 7−→ ϕ(b)/ϕ(g)n
61 / 142
A remarkable equivalence of categoriesIt turns out that every morphism of affine schemes is inducedby a ring homomorphism.
Proposition
There is a canonical isomorphism
Hom(Spec(A),Spec(B)) ' Hom(B,A).
Corollary
The functors A 7−→ Spec(A)
OX (X )←− [ X
define an arrow-reversing equivalence of categories betweenthe category of commutative rings and thecategory of affine schemes.
62 / 142
Adjointness of Spec and global sections
More generally:
Proposition
Let X be an arbitrary scheme, and let A be a ring. There is acanonical isomorphism
Hom(X ,Spec(A)) ' Hom(A, Γ(X )),
where Γ(X ) = OX (X ) is the ‘global sections’ functor.
63 / 142
Sum of schemesProposition
Let X1 and X2 be schemes. There exists a scheme X1∐
X2,called the sum of X1 and X2, together with morphismsXi → X1
∐X2 such that for every scheme Z
Hom(X1∐
X2,Z ) ' Hom(X1,Z )× Hom(X2,Z ),
i.e., every solid commutative diagram
Z
X1∐
X2
X1 X2
can be completed by a unique dashed morphism.
64 / 142
Proof.We reduce to affine schemes Xi = Spec(Ai). Then
X1∐
X2 = Spec(A1 × A2).
The underlying topological space of X1∐
X2 is a disjoint unionof the Xi .
65 / 142
Relative context
DefinitionLet us fix a scheme S.I An S-scheme, or a scheme over S is a morphism X → S.I A morphism of S-schemes is a diagram
X Y
S
66 / 142
Example
I Let k be a field (or even a ring) and S = Spec(k). Thecategory of affine S-schemes is equivalent to the categoryof k -algebras.
I If k is algebraically closed, and we consider only reducedfinitely generated k -algebras, the resulting category isessentially the category of affine algebraic varieties over k .
67 / 142
ProductsProposition
Let X1 and X2 be schemes over S. There exists a schemeX1 ×S X2, called the (fibre) product of X1 and X2 over S,together with S-morphisms X1 ×S X2 → Xi such thatfor every S-scheme Z
HomS(Z ,X1 ×S X2) ' HomS(Z1,X1)× HomS(Z ,X2),
i.e., every solid commutative diagram
Z
X1 ×S X2
X1 X2
Scan be completed by a unique dashed morphism.
68 / 142
Proof.We reduce to affine schemes Xi = Spec(Ai) over S = Spec(R).Then Ai are R-algebras and
X1 ×S X2 = Spec(A1 ⊗R A2).
69 / 142
Scheme-valued points
Definition
I Let X and T be schemes. The set of T -valued points of Xis the set
X (T ) = Hom(T ,X ).
I In a relative setting, suppose X , T are S-schemes. The setof T -valued points of X over S is the set
X (T )S = HomS(T ,X ).
70 / 142
Example
This notation is most commonly used as follows. Consider:I a system of polynomial equations fi = 0, i = 1, . . . ,m
defined over a field k , i.e., fi ∈ k [x1, . . . , xn];I A = k [x1, . . . , xn]/(f1, . . . , fn)
I and let K ⊇ k be a field extension.The associated scheme is X = Spec(A). Then
X (K )k = HomSpec(k)(Spec(K ),X )
= Homk (A,K )
' a ∈ K n : fi(a) = 0 for all i.
When k is algebraically closed, X (k) := X (k)k ⊆ kn is what wecalled an affine variety V (fi) at the start. The scheme Xcontains much more information.
71 / 142
Example ()
Suppose S is a scheme over a field k , and let X f→ S, Yg→ S
be two schemes over S (in particular, over k ). Then
(X ×S Y ) (k) = X (k)×S(k) Y (k)
= (x , y) : x ∈ X (k), y ∈ Y (k), f (x) = g(y).
72 / 142
Products vs topological products
Example
Zariski topology of the product is not the product topology, asshown in the following example.Let k be a field, then
A1 × A1 = A1 ×Spec(k) A1
= Spec(k [x1]⊗k k [x2]) ' Spec(k [x1, x2]) = A2.
The set of k -points A2(k) is the cartesian product
A1(k)× A1(k).
However, as a scheme, A2 has more points than the cartesiansquare of the set of points of A1.
73 / 142
Fibres of a morphism
DefinitionLet ϕ : X → S be a morphism, and let s ∈ S. There exists anatural morphism
Spec(k(s))→ S.
The fibre of ϕ over s is
Xs = X ×S Spec(k(s)).
RemarkXs should be thought of as ϕ−1(s), except that the abovedefinition gives it a structure of a k(s)-scheme.
74 / 142
Morphisms and families
Example
Consider R = k [z]→ A = k [x , y , z]/(y2 − x(x − 1)(x − z)) andthe corresponding morphism
ϕ : X = Spec(A)→ S = Spec(R).
Then, for s ∈ S corresponding to the ideal (z − λ), λ ∈ k ,
Xs = Xλ = Spec(k [x , y ]/(y2 − x(x − 1)(x − λ)))
so we can consider ϕ as a family of curves Xs with parameterss from S.
75 / 142
Reduction modulo p
Example
Consider Z→ A = Z[x , y ]/(y2 − x3 − x − 1) and thecorresponding morphism
ϕ : X = Spec(A)→ S = Spec(Z).
Then, for p ∈ S corresponding to the ideal (p) for a primeinteger p,
Xp = Xp = Spec(Fp[x , y ]/(y2 − x3 − x − 1)),
as a scheme over Fp = k(p), considered as a reduction of Xmodulo p.
76 / 142
Projective line as a compactification of A1
The picture for algebraic geometers:
P1
∞
P ′
P A1
I Think of P1 as the set of lines through a fixed point.I An equation ax + by + c = 0 describes a line if not both
a,b are 0, and what really matters is the ratio [a : b].
77 / 142
Riemann sphere
The picture for analysts:
<
=
∞
P ′
P
P1C
A1C
78 / 142
Projective space
Fix an algebraically closed field k .
DefinitionThe projective n-space over k is the set Pn
k of equivalenceclasses
[a0 : a1 : · · · : an]
of (n + 1)-tuples (a0,a1, . . . ,an) of elements of k , not all zero,under the equivalence relation
(a0, . . . ,an) ∼ (λa0, . . . , λan),
for all λ ∈ k \ 0.
79 / 142
Homogeneous polynomialsLet S = k [x0, . . . , xn].I For an arbitrary f ∈ S, P = [a0 : · · · : an] ∈ Pn, the
expressionf (P)
does not make sense.I For a homogeneous polynomial f ∈ S of degree d ,
f (λa0, . . . , λan) = λd f (a0, . . . ,an),
so it does make sense to consider whether
f (P) = 0 or f (P) 6= 0.
I It is beneficial to consider S as a graded ring
S =⊕d≥0
Sd ,
where Sd is the abelian group consisting of degree dhomogeneous polynomials.
80 / 142
Projective varieties
DefinitionLet T ⊆ S be a set of homogeneous polynamials. Let
V (T ) = P ∈ Pn : f (P) = 0 for all f ∈ T.
We writeD(f ) = Pn \ V (f ).
This has the expected properties and gives rise to the Zariskitopology on Pn.
DefinitionA projective algebraic variety is a subset of Pn of the form V (T ),together with the induced Zariski topology.
81 / 142
Covering the projective space with open affines
RemarkLet
Ui = D(xi) = [a0 : · · · : an] ∈ Pn : ai 6= 0 ⊆ Pn, i = 0, . . . ,n.
The maps ϕi : Ui → An defined by
ϕi([a0 : · · · : an]) =
(a0
ai, . . . ,
ai−1
ai,ai+1
ai, . . . ,
an
ai
).
are all homeomorphisms.Thus we can cover Pn by (n + 1) affine open subsets.
82 / 142
Example (from affine to projective curve)
Start with your favourite plane curve, e.g.,X = V (y2 − x3 − x − 1). Substitute y ← y/z, x ← x/z:
y2
z2 =x3
z3 +xz
+ 1.
Clear the denominators:
y2z = x3 + xz2 + z3.
This is a homogeneous equation of a projective curve X in P2,and X ∩ D(z) ' X .
83 / 142
Graded rings
Definition
I S is a graded ring ifI S =
⊕d≥0 Sd , Sd abelian subgroups;
I Sd · Se ⊆ Sd+e.I An element f ∈ Sd is homogeneous of degree d .I An ideal a in S is homogeneous if
a =⊕d≥0
(a ∩ Sd ),
i.e., if it is generated by homogeneous elements.
84 / 142
Projective schemesDefinitionLet S be a graded ring.I Let
S+ =⊕d>0
Sd E S.
I LetProj(S) = pE S : p prime, and S+ 6⊆ p.
I For a homogeneous ideal a, let
V+(a) = p ∈ Proj(S) : p ⊇ a.
I
D+(f ) = Proj(S) \ V+(f ).
As expected, V+ makes Proj(S) into a topological space.85 / 142
Structure sheaf on Proj(S)Notation: for p ∈ Proj(S), let S(p) be the ring of degree zeroelements in T−1S, where T is the multiplicative set ofhomogeneous elements in S \ p.
IntuitionIf a, f ∈ S are homogeneous of the same degree, then thefunction P 7→ a(P)/f (P) makes sense on D+(f ).
DefinitionFor U open in Proj(S),
O(U) =s : U →∐p∈U
S(p) | for each p ∈ U, s(p) ∈ S(p),
and for each p there is a nhood V 3 p, V ⊆ Uand homogeneous elements a, f of the same degreesuch that for all q ∈ V , f /∈ q, and s(q) = a/f in S(q).
86 / 142
Projective scheme is a scheme
Proposition
1. For p ∈ Proj(S), the stalk Op ' S(p).2. The sets D+(f ), for f ∈ S homogeneous, cover Proj(S),
and (D+(f ),O D+(f )
)' Spec(S(f )),
where S(f ) is the subring of elements of degree 0 in Sf .3. Proj(S) is a scheme.
Thus we obtained an example of a scheme which is not affine.
87 / 142
Global regular functions on projective varieties
Remark ()
The property 2. shows that
O(Proj(S)) = S0,
so the only global regular functions on Pn = Proj(k [x0, . . . , xn])are constant functions, since k [x0, . . . , xn]0 = k.The same statements holds for projective varieties.
Exercise: for k = C, deduce this from Liouville’s theorem.
88 / 142
Properties of schemes
I X is connected, or irreducible, if it is so topologically;I X is reduced, if for every open U, OX (U) has no nilpotents.I X is integral, if every OX (U) is an integral domain.
Lemma ()
X is integral iff it is reduced and irreducible.
89 / 142
Finiteness properties
I X is noetherian if it can be covered by finitely many openaffine Spec(Ai) with each Ai a noetherian ring;
I ϕ : X → Y is of finite type if there exists a covering of Y byopen affines Vi = Spec(Bi) such that for each i , ϕ−1(Vi)can be covered by finitely many open affinesUij = Spec(Aij) where each Aij is a finitely generatedBi -algebra;
I ϕ : X → Y is finite if Y can be covered by open affinesVi = Spec(Bi) such that for each i , ϕ−1(Vi) = Spec(Ai)with Ai is a Bi -algebra which is a finitely generatedBi -module.
90 / 142
PropernessDefinitionLet f : X → Y be a morphism. We say that f isI separated, if the diagonal ∆ is closed in X ×Y X ;I closed, if the image of any closed subset is closed;I universally closed, if every base change of it is closed, i.e.,
for every morphism Y ′ → Y , the corresponding morphism
X ×Y Y ′ → Y ′
is closed;I proper, if it is separated, of finite type and universally
closed.
ConventionHereafter, all schemes are separated!!!
91 / 142
Example
Finite morphisms are proper.
Prove this using the going up theorem of Cohen-Seidenberg:If B is an integral extension of A, then Spec(B)→ Spec(A) isonto.
92 / 142
Projective vars as algebraic analogues of compactmanifolds
Proposition
Projective varieties are proper (over k).
93 / 142
Images of morphisms
Example
Let Z = V (xy − 1), X = A1 and let π : Z → X be the projection(x , y) 7→ x .The image π(Z ) = A1 \ 0, so not closed.
Theorem (Chevalley)
Let f : X → Y be a morphism of schemes of finite type. Thenthe image of a constructible set is a constructible set (i.e., aBoolean combination of closed subsets).
94 / 142
Singularity, intuition via tangents on curves
Suppose we have a point P = (a,b) on a plane curve Xdefined by
f (x , y) = 0.
In analysis, the tangent to X at P is the line
∂f∂x
(P)(x − a) +∂f∂y
(P)(y − b) = 0.
I The partial derivatives of a polynomial make sense overany field or ring.
I In order for ‘tangent line’ to be defined, we need at leastone of ∂f
∂x (P), ∂f∂y (P) to be nonzero.
I Otherwise, the point P will be ‘singular’.
95 / 142
Example
The curve y2 = x3 has a singular point (0,0).
There are various types of singularities, this is a cusp.
96 / 142
Tangent space
DefinitionLet X ⊆ An be an irreducible affine variety, I = I(X ),P = (a1, . . . ,an) ∈ X . The tangent space TP(X ) to X at P is thesolution set of all linear equations
n∑i=1
∂f∂xi
(P)(xi − ai) = 0, f ∈ I.
It is enough to take f from a generating set of I.
Intuitive definition for varieties:We say that P is nonsigular on X if
dimk TP(X ) = dim X .
97 / 142
Derivations
DefinitionLet A be a ring, B an A-algebra, and M a module over B. AnA-derivation of B into M is a map
d : B → M
satisfying1. d is additive;2. d(bb′) = bd(b′) + b′d(b);3. d(a) = 0 for a ∈ A.
98 / 142
Module of relative differentials
DefinitionThe module of relative differential forms of B over A is aB-module ΩB/A together with an A-derivation d : B → ΩB/Asuch that: for any A-derivation d ′ : B → M, there exists a uniqueB-module homomorphism f : ΩB/A → M such that d ′ = f d :
B ΩB/A
M
d
d ′ ∃!f
99 / 142
Construction of ΩB/A
ΩB/A is obtained as a quotient of the free B-module generatedby symbols db : b ∈ B by the submodule generated byelements:
1. d(bb′)− bd(b′)− b′d(b), for b,b′ ∈ B ;2. da, for a ∈ A.
And the ‘universal’ derivation is just
d : b 7−→ (the coset of) db.
100 / 142
An intrinsic definition of the tangent space
LemmaLet X be an affine variety over an algebraically closed field k,P ∈ X. Let mP be the maximal ideal of OP . We haveisomorphisms
Derk (OP , k)∼→ Homk-linear(mP/m
2P , k)
∼→ TP(X ).
ThusΩOP/k ⊗OP k ' mP/m
2P .
Thus P is nonsingular iff dimk (mP/mnP) = dim(OP) iff ΩOP/k is a
free OP-module of rank dim(OP).
101 / 142
Nonsingularity
DefinitionA noetherian local ring (R,m) with residue field k = R/m isregular, if dimk (m/m2) = dim(R).
By Nakayama’s lemma, this is equivalent to m havingdim(R) generators.
Definition
I A noetherian scheme X is regular, or nonsingular at x , ifOx is a regular local ring.
I X is regular/nonsingular if it is so at every point x ∈ X .
102 / 142
Sheaves of differentials; regularity vs smoothnessLet ϕ : X → Y be a morphism. There exists a sheaf of relativedifferentials ΩX/Y on X and a sheaf morphism d : OX → ΩX/Ysuch that:if U = Spec(A) ⊆ Y and V = Spec(B) ⊆ X are open affinesuch that f (V ) ⊆ U, then ΩX/Y (V ) = ΩB/A.
Proposition
Let X be an irreducible scheme of finite type over analgebraically closed field k. Then X is regular over k iff ΩX/k isa locally free sheaf of rank dim(X ), i.e., every point has anopen neighbourhood U such that
ΩX/k U ' (OX U)dim(X).
Over non-algebraically closed field the latter is associatedwith a notion of smoothness.
103 / 142
Generic non-singularity
Corollary
If X is a variety over a field k of characteristic 0, then there isan open dense subset U of X which is nonsingular.
Example
Funny things can happen in characteristic p > 0; think of thescheme defined by xp + yp = 1.
104 / 142
DVR’s
DefinitionLet K be a field. A discrete valuation of K is a mapv : K \ 0 → Z such that
1. v(xy) = v(x) + v(y);2. v(x + y) ≥ min(v(x), v(y)).
Then:I R = x ∈ K : v(x) ≥ 0 ∪ 0 is a subring of K , called the
valuation ring;I m = x ∈ K : v(x) > 0 ∪ 0 is an ideal in R, and (R,m)
is a local ring.
DefinitionA valuation ring is an integral domain R which the valuation ringof some valuation of Fract(R).
105 / 142
Characterisations of DVR’s
FactLet (R,m) be a noetherian local domain of dimension 1. TFAE:
1. R is a DVR;2. R is integrally closed;3. R is a regular local ring;4. m is a principal ideal.
106 / 142
RemarkLet X be a nonsingular curve, x ∈ X. Then Ox is a regular localring of dimension 1, and thus a DVR.
A uniformiser at x is a generator of mx .
107 / 142
Dedekind domains
FactLet R be an integral domain which is not a field. TFAE:
1. every nonzero proper ideal factors into primes;2. R is noetherian, and the localisation at every maximal ideal
is a DVR;3. R is an integrally closed noetherian domain of dimension 1.
DefinitionR is a Dedekind domain if it satisfies (any of) the aboveconditions.
RemarkIf X is a nonsingular curve, then O(X ) is a Dedekind domain.
108 / 142
Divisors
DefinitionLet X be an irreducibe nonsingular curve over an algebraicallyclosed field k .I A Weil divisor is an element of the free abelian group DivX
generated by the (closed) points of X , i.e., it is a formalinteger combination of points of X .
I A divisor D =∑
i nixi is effective, denoted D ≥ 0 if allni ≥ 0.
109 / 142
Principal divisorsDefinitionLet X be an integral nonsingular curve over an algebraicallyclosed field k , and let K = k(X ) = Oξ = lim−→U open
OX (U) be itsfunction field (where ξ is the generic point of X ), which we thinkof as the field of ‘rational functions’ on X .For f ∈ K×, we let the divisor (f ) of f on X be
(f ) =∑
x∈X 0
vx (f ) · x ,
where vx is the valuation in Ox . Any divisor which is equal tothe divisor of a function is called a principal divisor.
RemarkNote this is a divisor: if f is represented as fU ∈ OX (U) onsome open U, and thus (f ) is ‘supported’ on V (fU) ∪ X \ U,which is a proper closed subset of X and it is thus finite.
110 / 142
Remarkf 7→ (f ) is a homomorphism K× → DivX whose image is thesubgroup of principal divisors.
DefinitionFor a divisor D =
∑i nixi , we define the degree of D as
deg(D) =∑
i
ni ,
making deg into a homomorphism DivX → Z.
111 / 142
Divisor class group
DefinitionLet X be a non-singular difference curve over k .I Two divisors D,D′ ∈ DivX are linearly equivalent, written
D ∼ D′, if D − D′ is a principal divisor.I The divisor class group ClX is the quotient of DivX by the
subgroup of principal divisors.
112 / 142
Ramification
DefinitionLet ϕ : X → Y be a morphism of nonsingular curves, y ∈ Y andx ∈ X with π(x) = y .The ramification index of ϕ at x is
ex (ϕ) = vx (ϕ]ty ),
where ϕ] is the local morphism Oy → Ox induced by ϕ andty is a uniformiser at y , i.e., my = (ty ).When ϕ is finite, we can define a morphism ϕ∗ : DivY → DivXby extending the rule
ϕ∗(y) =∑
ϕ(x)=y
ex (ϕ) · x
for prime divisors y ∈ Y by linearity to DivY .
113 / 142
Preservation of multiplicityTheoremLet ϕ : X → Y be a morphism of nonsingular projective curveswith ϕ(X ) = Y, then degϕ = deg(ϕ∗(y)) for any point y ∈ Y.
Proof reduces to the Chinese Remainder Theorem.
114 / 142
The number of poles equals the number of zeroes
Corollary
The degree of a principal divisor on a nonsingular projectivecurve equals 0.
Proof.Any f ∈ k(X ) defines a morphism f : X → P1. Then
deg((f )) = deg(f ∗(0))− deg(f ∗(∞)) = deg(f )− deg(f ) = 0.
RemarkHence deg : Cl(X )→ Z is well-defined.
115 / 142
Bezout’s theorem
Theorem (Bezout)
Let X ⊆ Pn be a nonsingular projective curve, and letH = V+(f ) ⊆ Pn be the hypersurface defined by ahomogeneous polynomial f . Then, writing
X .H =∑
x∈X∩H
i(x ; X ,H)x := (f ),
we have thatdeg(X .H) = deg(X ) deg(f ),
where deg(X ) is the maximal number of points of intersectionof X with a hyperplane in Pn (which does not contain acomponent of X).
116 / 142
Proof.Let d = deg(f ). For any linear form l , h = f/ld ∈ k(X ), so
deg((f )) = deg((ld )) + deg((h)) = d deg(l) + 0 = d deg(X ).
117 / 142
Elliptic curvesLet E be a nonsingular projective plane cubic, and pick a pointo ∈ E . For points p,q ∈ E , let p ∗ q be the unique point suchthat, writing L for the line pq and using Bezout,E .L = p + q + p ∗ q. We define
p ⊕ q = o ∗ (p ∗ q).
Example (E . . . y2z = x3 − 2xz2, o =∞ := [0 : 1 : 0])
x
yy2 = x3 − 2x
o =∞
p•q •
•p ∗ q
•p ⊕ q118 / 142
Proposition
Let (E ,o) be an elliptic curve, i.e., a nonsingular projectivecubic over k. Then (E(k),⊕) is an abelian group.
Proof.Only the associativity of ⊕ needs checking. For a fun proofusing nothing other than Bezout’s Theorem see Fulton’s Alg.Curves.
119 / 142
Aside on algebraic groups
DefinitionA group variety over S = Spec(k) is a variety X π→ S togetherwith a section e : S → X (identity), and morphismsµ : X ×S X → X (group operation) and ρ : X → X (inverse)such that
1. µ (id × ρ) = e π : X → X ;2. µ (µ× id) = µ (id × µ) : X × X × X → X .
Clearly, for a field K extending k , X (K ) is a group.
120 / 142
Examples of algebraic groups
Examples
1. Additive group Ga = A1k . Multiplicative group
Gm = Spec(k [x , x−1]).2. SL2(k) = (a,b, c,d) : ad − bc = 1.
ρ(a,b, c,d) = (d ,−b,−c,a) etc.3. GL2(k) = Spec(k [a,b, c,d ,1/(ad − bc)]).
121 / 142
Elliptic curves are abelian varieties
Proposition
Let (E ,o) be an elliptic curve. Then (E ,⊕) is a group variety.
In other words, the operations ⊕ : E × E → E and : E → Eare morphisms.
DefinitionAn abelian variety is a connected and proper group variety (itfollows that the operation is commutative, hence the name).
Thus, elliptic curves are examples of abelian varieties.
122 / 142
The canonical divisor
DefinitionLet X be an integral non-singular projective curve over k . ThenΩX/k is a locally free sheaf of rank 1, and pick a non-zero globalsection ω ∈ ΩX/k (X ). For x ∈ X , let t be the uniformiser at x ,and let f ∈ k(X ) be such that ω = f dt . Define
vx (ω) = vx (f ),
and the resulting canonical divisor
W =∑
x
vx (ω)x .
The divisor W ′ of a different ω′ ∈ ΩX/k (X ) is linearly equivalentto W , W ′ ∼W , and thus W uniquely determines a canonicalclass KX in ClX .
123 / 142
Example
[Canonical divisor of an elliptic curve]
124 / 142
Complete linear systemsDefinitionLet D be a divisor on X , and write
L(D) = f ∈ k(X ) : (f ) + D ≥ 0.
A theorem of Riemann shows that these are finite dimensionalvector spaces over k , and let
l(D) = dim L(D).
Remarkf and f ′ define the same divisor iff f ′ = λf , for some λ 6= 0, sowe have a bijection
effective divisors ∼ D ↔ P(L(D)).
125 / 142
Riemann-Roch Theorem
DefinitionThe genus of a curve X is l(KX ).
Theorem (Riemann-Roch)
Let D be a divisor on a projective nonsingular curve X of genusg over an algebraically closed field k. Then
l(D)− l(KX − D) = deg(D) + 1− g.
In particular, deg(KX ) = 2g − 2.
126 / 142
The zeta function
DefinitionLet X be a ‘variety’ over a finite field k = Fq. Its zeta function isthe formal power series
Z (X/Fq,T ) = exp
∑n≥1
|X (Fqn )|n
T n
.
127 / 142
Examples
I Let X = ANFq
. We have |ANFq
(Fqn )| = qnN , so
Z (ANFq,T ) = exp
∑n≥1
(qNT )n
n
=1
1− qNT.
I For X = PNFq
,
PNFq
(Fqn ) =qn(N+1) − 1
qn − 1= 1 + qn + q2n + · · ·+ qNn, so
Z (PNFq/Fq,T ) = exp
∑n≥1
T n
n
N∑j=0
qnj
=N∏
j=0
Z (AjFq/Fq,T )
=N∏
j=0
11− qjT
.
128 / 142
Frobenius
Suppose X is over Fq, consider the algebraic closure Fq of Fq,and the Frobenius automorphism
Fq : Fq → Fq, Fq(x) = xq.
Then Fq acts on X (Fq) = Hom(Spec(Fq),X ) by precomposingwith aFq.
Intuitively, if X is affine in AN , then
Fq(x1, . . . , xN) = (xq1 , . . . , x
qN).
Remark
X (Fqn ) = Fix(F nq ).
129 / 142
Points vs geometric points
RemarkA closed point x ∈ X corresponds to an Fq-orbit of anx ∈ X (Fq), and
[k(x) : Fq] = |orbit of x| = minn : x ∈ X (Fqn ).
DefinitionFor a closed point x ∈ X , let
deg(x) = [k(x) : Fq], Nx = qdeg(x).
130 / 142
Comparison with the Riemann zeta
Recall Riemann’s definition:
ζ(s) =∑n≥1
n−s =∏
p∈SpecmZ
(1− p−s)−1.
Lemma
Z (X/Fq,T ) =∏
x∈X 0
(1− T deg(x))−1,
i.e., after a variable change T ← q−s,
Z (X/Fq,q−s) =∏
x∈X 0
(1− Nx−s)−1.
131 / 142
Proof.Exercise upon remarking that
|X (Fqn )| =∑r |n
r · |x ∈ X 0 : deg(x) = r|.
132 / 142
The Weil ConjecturesLet X be a smooth projective variety of dimension d overk = Fq, Z (T ) := Z (X/k ,T ). Then
1. Rationality. Z (T ) is a rational function.2. Functional equation.
Z (1
qdT) = ±Tχqχ/2Z (T ),
where χ is the ‘Euler characteristic’ of X .3. Riemann hypothesis.
Z (T ) =P1(T )P3(T ) · · ·P2d−1(T )
P0(T )P2(T ) · · ·P2d (T ),
where each Pi(T ) has integral coefficients and constantterm 1, and
Pi(T ) =∏
j
(1− αijT ),
where αij are algebraic integers with |αij | = qi/2. Thedegree of Pi is the ‘i-th Betti number’ of X .
133 / 142
I The use of ‘Euler characteristic’ and ‘Betti numbers’implies that the arithmetical situation is controlled by theclassical geometry of X .
I History of proof: Dwork, Grothendieck-Artin, Deligne.I We shall sketch the rationality for curves.
134 / 142
Divisors over non-algebraically closed base field
DefinitionLet X be a curve over k .I Div(X ) is the free abelian group generated by the closed
points of X .I For D =
∑i nixi ∈ Div(X ), let
deg(D) =∑
i
ni deg(xi).
I Write Div(n) = D ∈ Div(X ) : deg(D) = n andCl(n) = Div(n)/∼.
135 / 142
Structure of divisor class groups
Using Riemann-Roch, if deg(D) > 2g − 2, thendeg(KX − D) < 0 so l(KX − D) = 0 and thus
l(D) = deg(D) + 1− g.
Therefore, for n > 2g − 2, the number En of effective divisors ofdegree n is
∞ > En =∑
D∈Cl(n)
ql(D) − 1q − 1
=∑
D∈Cl(n)
qn+1−g − 1q − 1
= |Cl(n)|qn+1−g − 1
q − 1.
In particular, |Cl(n)| <∞.
136 / 142
Structure of divisor class groups
Suppose the image of deg : Div(X )→ Z is dZ (we will see laterthat d = 1). Choosing some D0 ∈ Div(d) defines anisomorphism
Cl(n)∼−→ Cl(n + d)
D 7−→ D0 + D,
and therefore
|Cl(n)| =
J if d |n0 otherwise,
where J = |Cl(0)| is the number of rational points on theJacobian of X .NB d |2g − 2 since deg(KX ) = 2g − 2.
137 / 142
Rationality of zeta for curves
Z (X/Fq,T ) =∏
x∈X 0
(1− T deg(x))−1 =∑D≥0
T deg(D) =∑n≥0
EnT n
=
2g−2∑n=0d |n
T n∑
D∈Cl(n)
ql(D) − 1q − 1
+∞∑
n=2g−2+dd |n
T nJqn+1−g
q − 1
= Q(T ) +J
q − 1T 2g−2+d
[qg−1+d
1− (qT )d −1
1− T d
],
so Z (X/Fq,T ) is a rational function in T d with first order polesat T = ξ, T = ξ
q for ξd = 1.
138 / 142
Lemma (Extension of scalars)
Z (X ×Fq Fqr /Fqr ,T d ) =∏ξr =1
Z (X/Fq, ξT ).
Proposition
d = 1.
Proof.By an analogous argument, Z (X ×Fq Fqd/Fqd ,T d ) has a firstorder pole at T = 1. Using extension of scalars and the factthat Z (X/Fq,T ) is a function of T d , we get
Z (X ×Fq Fqd/Fqd ,T d ) =∏ξd =1
Z (X/Fq, ξT ) = Z (X/Fq,T )d .
Comparing poles, we conclude d = 1.139 / 142
Functional equation for curves
RemarkBy inspecting the above calculation of Z (X/Fq,T ), usingRiemann-Roch, one can deduce the functional equation
Z (X/Fq,1
qT) = q1−gT 2−2gZ (X/Fq,T ).
140 / 142
Cohomological interpretation of Weil conjectures
Let X be a variety of dimension d over k = Fq, X = X ×k k andlet F : X → X be the Frobenius morphism. Fix a primel 6= p = char(k). There exist l-adic étale cohomology groups(with compact support)
H i(X ) = H ic(X ,Ql), i = 0, . . . ,2d
which are finite dimensional vector spaces over Qlso that F induces vector space morphisms F ∗ : H i(X )→ H i(X )and we have a Lefschetz fixed-point formula
|X (Fqn )| = |Fix(F n)| =2d∑i=0
(−1)i tr(F ∗n|H i(X )).
141 / 142
Weil rationality using cohomology
Z (X ,T ) = exp
∑n≥1
T n
n
2d∑i=0
(−1)i tr(F ∗n|H i(X ))
=
2d∏i=0
exp
∑n≥1
tr(F ∗n|H i(X ))T n
n
(−1)i
=2d∏i=0
[det(1− F ∗T |H i(X ))
](−1)i
an alternating product of the characteristic polynomials of theFrobenius on cohomology.
142 / 142